N/p \quad \text{if } \ 1 < p < N \\
r = 1 \quad \text{if } \ p > N \\
r > p \quad \text{if } \ p = N
\end{gathered} \label{e1}
\end{equation}
We assume that $\mathop{\rm meas}(\Omega^{+}) \neq 0 $, where
$\Omega^{+}=\{ x \in \Omega /m(x)>0 \}$. We consider the
nonlinear eigenvalue problem
\begin{equation}
\begin{gathered}
- \Delta_p u =\lambda m(x)|u|^{p-2}u \quad \text{in } \Omega,\\
{{\partial u}\over {\partial\nu}} =0 \quad \text{on } \partial\Omega,
\end{gathered}\label{e2}
\end{equation}
where $\Delta_p u = \mathop{\rm div}(|\nabla u |^{p-2}\nabla u)$
is the p-Laplacian operator.
The goal of this work is to
prove the existence of a sequence of non trivial eigenvalues for
the problem \eqref{e2}, and we prove the simplicity, isolation
and monotonicity with respect to the weight of the first
eigenvalue $\lambda_1$ defined by
\begin{equation}
\lambda_1=\inf \{ \|\nabla u\|_p^{p}: u \in
W^{1,p}(\Omega) \text{ and } \int_\Omega m(x)|u|^{p}dx=1
\}. \label{e3}
\end{equation}
The semilinear elliptic problems has been treated by many
authors; see, e.g. \cite{b1,h1,h2,o1}
and the references therein.
In the case of bounded weight: Anane \cite{a1} with Dirichlet
boundary
conditions and Dakkak \cite{d1} with Neumann boundary conditions,
proved the existence, simplicity and isolation of the first
eigenvalue. Cuesta \cite{c2} (for the p-Laplacian) and in
Touzani \cite{t1} (for the $A_p$-Laplacian:
$A_{p}u={\Sigma}_{i,j=1}^{N}{{\partial}\over{\partial
x_i}}(|\nabla u|_{a}^{p-2}a_{ij}(x){{\partial u}\over{\partial
x_j}})$, $|\xi|_{a}^{2}= \Sigma_{i,j=1}^{N}a_{ij}(x)\xi_i\xi_j$)
studied the above properties in the case of Dirichlet problem with
weight in $L^r(\Omega)$, with $r$ satisfying
\eqref{e1}. Cuesta \cite{c2} showed these results for any $r$ satisfying
the conditions \eqref{e1}, by using Harnack's inequality and Picone's
identity. However, in \cite{t1}, the isolation was establisehd
with some appropriate condition on $r$ ($r> Np'$) in order to use
the regularity results established by Di Benedetto \cite{d2}. We
will try to adapt these results to the case of Neumann boundary
conditions.
This paper is organized as follows. In section 2, we
recall some definitions and results that we will use later. In
section 3, we prove that the problem \eqref{e2} has a
sequence of eigenvalues, by using a perturbation of
the initial problem (\cite{c2}), and then by applying the
Ljusternik-Schnirelmann theory (\cite{b2,b3}) to the perturbed
problem. In section 4, we show simplicity, isolation and
monotonicity of the first eigenvalue $ \lambda_1$.
\section{Preliminaries}
Note by $W^{1,p}(\Omega)$ the Sobolev space with norm
$\|.\|_{1,p}= (\|.\|_{p}^{p} + \|\nabla(.)\|_{p}^{p})^{1/p}$,
where $\|.\|_p$ is the $L^p$-norm.
We say that $\lambda \in \mathbb{R} $ is an eigenvalue of problem
\eqref{e2} if there exists $ u \in W^{1,p}(\Omega)
\setminus \{0\} $ such that
\begin{equation}
\int_\Omega |\nabla u|^{p-2} \nabla u
\nabla \varphi \,dx = \lambda \int_\Omega m(x) |u|^{p-2} u
\varphi \,dx \hspace{6mm}{ \forall} \varphi \in W^{1,p}(\Omega).
\label{e4}
\end{equation}
\begin{theorem}[\cite{a1}] \label{thm2.1}
Let $v>0$ and $u\geq 0$ be two continuous functions in $\Omega$,
differentiable a.e, and
\begin{gather*}
L(u,v) =|\nabla u|^{p} +(p-1){{u^{p}}\over{v^{p}}}|\nabla v|^{p} -
p {{u^{p-1}}\over{v^{p-1}}}|\nabla v|^{p-2}\nabla v
\nabla u ,\\
R(u,v) =|\nabla u|^{p} - |\nabla v|^{p-2} \nabla
({{u^{p}}\over{v^{p-1}}}) \nabla v .
\end{gather*}
Then we have
\begin{itemize}
\item[(i)] L(u,v) =R(u,v) \item[(ii)] $L(u,v)\geq 0 $ a.e. in
$\Omega$ \item[(iii)] $L(u,v)=0$ a.e. in $\Omega$ if and only if
there exists $ k \in \mathbb{R}$ such that $u=k v$ .
\end{itemize}
\end{theorem}
\begin{proposition} \label{prop2.1}
Let $u \in W^{1,p}(\Omega) $ be an eigenfunction associated to
$\lambda$ then
\begin{itemize}
\item[(i)] $u \in L^{\infty}(\Omega)$ \item[(ii)] $u$ is locally
H\"{o}lder continuous; i.e., there exists
$\alpha=\alpha(p,N,\|\lambda m\|_r )$ in $]0,1[$ such that for
each $\Omega'\subset\Omega$, there exists $C=C(p,N,\|\lambda
m\|_s,\mathop{\rm dist}(\Omega',\partial\Omega))$ such that
$$
|u(x)-u(y)|\leq C\|u\|_\infty |x-y|^{\alpha} \quad \forall x,y
\in \Omega'.
$$
\end{itemize}
\end{proposition}
The proof of (i) can be found in
\cite[propositions 1.2 and 1.3]{g1}, and of (ii)
in \cite[theorem 8]{s1}.
\begin{proposition} \label{prop2.2}
Let $u \in W^{1,p}(\Omega) $
be a nonnegative weak solution of \eqref{e2}, then either
$u\equiv 0$ or $ u(x) > 0$ for all $ x \in \Omega$.
\end{proposition}
The proof is a direct consequence of
Harnack's inequality (see \cite[theorem 5, 6, 9]{s2}).
\section{Existence of solutions}
In this section, we establish the existence of solutions by using
a perturbation of problem \eqref{e2} in order to use the general
theory of nonlinear eigenvalue problems. So, let us consider the
perturbed problem
\begin{equation}
\begin{gathered}
- \Delta_p u +\varepsilon |u|^{p-2}u=\lambda m(x)|u|^{p-2}u \quad
\text{in} \ \Omega,\\
{{\partial u}\over {\partial\nu}} =0 \quad \textrm{on} \
\partial\Omega,
\end{gathered}
\label{e5}
\end{equation}
where $\varepsilon$ is enough small $(00$. We first show that problem \eqref{e5} has at least
one sequence of eigenvalues, and deduce the solutions of problem
\eqref{e2} when $\varepsilon $ tends to 0.
Let $X=W^{1,p}(\Omega)$ , $G_\varepsilon(u) = {{1}\over{p}}
\|\nabla u\|_p^{p} +{{ \varepsilon} \over {p}}\|u\|_p^{p} $, and
$F(u) = {{1}\over{p}}\int_\Omega m(x) |u|^{p}dx $. It is well
known that $F$ and $G_\varepsilon$ are differentiable \cite{d1}.
The problem \eqref{e5} is equivalent to the problem $
G_\varepsilon'(u) = \lambda F'(u)$. Let us consider the functional
$ \Phi_\varepsilon:W^{1,p}(\Omega) \to \mathbb{R}$ defined by
$\Phi_\varepsilon(v)= (G_\varepsilon(v))^{2} - F(v)$.
\begin{lemma} \label{lm3.1}
The eigenvalues and eigenfunctions associated to the problem
\eqref{e5} are entirely determined by a non trivial critical
values of $\Phi_\varepsilon$.
\end{lemma}
\begin{proof} Let $u \not\equiv 0 $ be a critical point of
$\Phi_\varepsilon$ associated with a critical value
$c_\varepsilon$ then $\Phi_\varepsilon(u) =c_\varepsilon $ and
$\Phi_\varepsilon'(u) = 0 $, i.e $c_\varepsilon =
-(G_\varepsilon(u))^{2} < 0 $ and $\langle \Phi_\varepsilon'(u),v
\rangle = {{1} \over {2\sqrt{-c_\varepsilon}}} \langle F'(u),v
\rangle$ for any $v \in C_c^{\infty}(\Omega)$. Thus we deduce
that $ \lambda = {{1} \over {2\sqrt{-c_\varepsilon}}}$ is a
positive eigenvalue of \eqref{e5} and $u$ is its associated
eigenfunction.
Conversely, let $ (u\not\equiv 0 , \lambda)$ be a solution of
\eqref{e5}, then
for every $\beta \in {\mathbb{R}}^{\ast}, \beta u $ is also an
eigenfunction associated to $\lambda$. In particular for $\beta
=( {{1} \over {2 \lambda G_\varepsilon(u)}})^{{1}\over{p}}$,
$ v =(2 \lambda G_\varepsilon(u))^{-{{1}\over{p}}} u $ is an
eigenfunction associated to $\lambda =
{{1}\over{2\sqrt{-c_\varepsilon}}}$, thus $v$ is a critical point
associated to the critical value $c_\varepsilon ={-{{1}\over{4
\lambda^{2}}}} $.
\end{proof}
Let us now consider the sequence
\begin{equation}
c_{n,\varepsilon }=\inf_{K \in
A_n} \sup_{v \in K} \Phi_\varepsilon(v), \label{e6}
\end{equation}
where $ A_n = \{ K \subset W^{1,p}(\Omega) : K \text{ is compact
symmetric and } \gamma(K) \geq n \}$, $n \geq 1$.
\begin{theorem} \label{thm3.1}
The values $c_{n,\varepsilon} $ defined by \eqref{e6} are the
critical values of $\Phi_\varepsilon $, moreover $
c_{n,\varepsilon} < 0 $ for $n \geq 1$ and $\lim_{n \to
\infty}{c_{n,\varepsilon}} = 0 $.
\end{theorem}
\begin{proof} The proof of this theorem is based on the
fundamental theorem of multiplicity \cite{c1} and the
approximation of Sobolev imbedding by operators of finite rank. We
first show that for all $n\geq 1$, $c_{n,\varepsilon}$ is a
critical value of $\Phi_\varepsilon $ and $ c_{n,\varepsilon}<0 $.
Since $\phi_\varepsilon $ is even and is $ C^{1}$ on
$W^{1,p}(\Omega)$, then the result follows from the fundamental
theorem of multiplicity if $ \Phi_\varepsilon $ satisfies the
following conditions:
\begin{itemize}
\item[(i)] $ \Phi_\varepsilon $ is bounded below
\item[(ii)] $\Phi_\varepsilon $ verify the Palais Smale condition
(PS).
\item[(iii)] for all $n \geq 1$, there exists a compact symmetric
subset $K$ such that $\gamma(K) = n$ and $\sup_{v \in
K}\{\Phi_\varepsilon(v)\} < 0 $.
\end{itemize}
Let us verify assertion (i): Let us take $\varepsilon$ fixed ($
0 0$ , $k_2 > 0$ and $k_3>0$ such that:
$$
\int_\Omega m|u|^{p} dx \leq
\begin{cases} k_1\|m\|_r\|u\|_{1,p}^{p} & \text{if } 1

N
\end{cases}
$$
In addition, note that $ \|u\|_{1,p,\varepsilon} = (\|\nabla
u\|_{p}^{p} + \varepsilon \|u\|_p^{p})^{{1}\over{p}}$ defines a
norm on $W^{1,p}(\Omega)$ equivalent to the usual norm on
$W^{1,p}(\Omega)$. It is easy to see that there exists
$c_1>0$, $c_2>0 $ and $c_3>0$ such that for $\Phi_\varepsilon$,
we have the following inequalities:
$$
\Phi_\varepsilon(u) \geq
\begin{cases} {{c_1}\over{p^{2}}}
\|u\|_{1,p}^{p}(\varepsilon^{2} \|u\|_{1,p}^{p} - p \|m\|_r)
& \text{if } 1